The conclusion is correct. But I don't see why [itex]\frac{ \log ^{4}(n)}{n} \rightarrow 0[/itex] is any more obvious than [itex]\frac{ \log ^{5}(n)}{n} \rightarrow 0[/itex]. Why don't you keep applying l'Hopital until the log disappears?

The conclusion is correct. But I don't see why [itex]\frac{ \log ^{4}(n)}{n} \rightarrow 0[/itex] is any more obvious than [itex]\frac{ \log ^{5}(n)}{n} \rightarrow 0[/itex]. Why don't you keep applying l'Hopital until the log disappears?